The main difficulties of our problem come from the delayed term and the non-Lipschitz generator.
For the former one, we need to overcome the trouble from the infinite-dimensional of the space \(C([0,t];\mathbb {R}^n)\)  in the verification that the optimal value function satisfies the HJB equation. We manage to do it with the help of the semigroup property of generator in the infinite-dimensional setting, by borrowing ideas from Fuhrman–Masiero–Tessitore [1]} and Chen–Wu [2]}.
The non-Lipschitz generator brings troubles in the establishment of the relationship between the value function and its associated HJB equation. We conquer it by applying the techniques for the solvability of BSDE with monotonic coefficients in e.g. Pardoux [3]} and the stability of viscosity solutions in Fleming-Soner [4]}.
